No, he doesn't. The Tokyo Revengers manga doesn't have Takemichi's death. Instead, he plays a crucial role in the unfolding events and continues to be an important part of the story.
In Takemichi x Mikey fanfiction, a common theme could be their growth together. Takemichi's time - leaping ability can create really interesting storylines in relation to Mikey. For example, Takemichi might use his power to save Mikey from a dark future, and in the process, their relationship evolves. It's also possible that the fanfiction shows their shared dreams and how they strive to achieve them side by side.
Takemichi's time - leaping ability in the Tokyo Revengers real story is quite unique. It doesn't seem to follow a set pattern like in some other time - travel stories. As I said before, emotions play a big role. But also, it's not something he can fully control. Sometimes it happens unexpectedly. For example, when he's in a dangerous situation and his mind is focused on going back to change things, he is suddenly transported back. This ability is what drives the whole plot as he uses it to try and make a better future for his friends and Hinata by changing the past.
The popularity also stems from the fans' love for the Tokyo Revengers world in general. Since Takemichi and Mikey are central characters, fans are eager to explore their relationship further. Fanfiction allows them to do that, whether it's imagining different timelines, different interactions, or just exploring their emotional connection on a deeper level.